In this chapter we will discuss the effect of shear viscosity on evolution of fluid, p T

Transcription

1 Chapter 3 Shear viscous evolution In this chapter we will discuss the effect of shear viscosity on evolution of fluid, p T spectra, and elliptic flow (v ) of pions using a +1D relativistic viscous hydrodynamics simulation (details of which are given in chapter-). To study the shear viscous evolution we have to solve the evolution equation for the energy-momentum tensor and the relaxation equations for shear stress tensor. In particular, the energy momentum conservation equation in presence of only shear viscosity has the following form[3, 11], τ Tττ + x ( vx Tττ ) + y ( vy Tττ ) = [ P +τ π ηη], τ T τx + x ( vx T τx) + y ( vy T τx) = x [ P + π xx v x π τx] y [ π yx v y π τx ], T ( τy τ + T τy) ( x vx + T τy) [ y vy = y P + π yy v y π τy] x [ π xy v x π τy ]. (3.1) Where Ãmn τa mn, P τp, vx T τx /T ττ, and v y T τy /T ττ. The relaxation equation for the component π xx of the shear stress π µν is τ π xx +v x x π xx +v y y π xx = 1 τ π γ (πxx ησ xx ) 1 γ Ixx 1. (3.) 7

2 79 The evolution equations for π yy and π xy are of similar form (see chapter-). We will denote the term I µν 1 (π λµ u ν +π λν u µ )Du λ as the R-term. The effect of inclusion of this term on the fluid evolution as well as on the spectra and v of pions will be discussed. It has been pointed out in [7] that the inclusion of R-term is important. The R-term ensures that throughout the evolution π µν remains traceless and it also ensures that π µν u µ =. Shear viscous evolution without the R-term is named as simplified Israel- Stewart equation in this chapter. Initial conditions : The simulations have been done for Au-Au collisions with initial central energy density ǫ = 3 GeV/fm 3 and for Pb-Pb collisions with ǫ = 1 GeV/fm 3. Thesevaluesofǫ approximatelycorrespondstotheenergydensitiesachieved attoprhicenergy( s NN =GeV)andLHCenergy( s NN =.7TeV)respectively. The two component Glauber model initialization is used for calculating the transverse energy density profile. The impact parameter for simulating the heavy ion collisions is taken as b=7. fm. Initial time is. fm for both RHIC and LHC. The initial transverse velocity of the fluid is assumed to be zero (v x (x,y) = v y (x,y) = ). At the initial time τ, the values of the independent component of shear stresses π xx,π yy, and π xy are set to the corresponding Navier-Stokes values for a boost invariant expansion as, π xx = ησ xx = η 3τ, π yy = ησ yy = η 3τ, π xy = ησ xy =. The σ µν s are already defined in chapter-.

3 Equation of State : We have used lattice+hrg equation of state with crossover transition at T co =175 MeV as discussed in the previous chapter. The low temperature phase of the EoS is modeled by hadronic resonance gas, containing all the resonances with mass M res.5 GeV. The high temperature phase is a parametrization of the recent lattice QCD calculation [1]. Entropy density of the two phases were smoothly joined at T = T co =175 MeV by a smooth step like function. Shear viscous coefficient : The simulations have been carried out for four different input values of η/s=. (ideal),. (KSS bound []),.1, and.1. They are considered to be independent of temperature. Shear relaxation time : The default value of the shear relaxation time τ π for the simulation results presented in this chapter is.5 (3η/p). The effect of varying τ π will be discussed later in this chapter. Freezeout : The freezeout temperature is set to T fo =13 MeV. The effect of a higher freezeout temperature T fo =1 MeV has also been studied. The implementation of the shear viscous correction to the freezeout distribution function will be discussed in the last subsection of this chapter.

4 1 3.1 Temporal evolution of fluid In presence of shear viscosity, the thermodynamic pressure is modified. The tracelessness of shear stress tensor π µν, along with the assumption of longitudinal boost invariance ensures that the π xx and π yy components are positive at the initial time of the fluid evolution. Consequently, for the same thermodynamic conditions, the effective pressure is larger (see equation 3.1) in the transverse direction compared to the ideal fluid. It is then important to have some idea how various components of shear viscous stress π µν evolves in space-time. In the next section, we will first discuss the evolution of the independent components of π µν which are π xx, π yy, and π xy according to our convention. We will also show in detail the evolution of average values of all the seven components of π µν as a function of time Evolution of shear-stress The temporal evolution of π xx (x,y), π yy (x,y), and π xy (x,y) for two different relaxation time τ π =.5 (3η/p) and τ π =.1 (3η/p) are shown in figures 3.1 and 3. respectively. Results are shown in one quadrant only, values in the other quadrants can be obtained by using refelction symmetry. For impact parameter b=7. fm collision, the reaction zone is elliptical, the π xx (x,y), π yy (x,y) and π xy (x,y) also reflects this elliptical shape. The magnitude of π xx and π yy is observed to decrease with the increase in distance from the center of the reaction zone. The values also decrease with increase in τ. The value of π xx and π yy reduces by almost two orders of magnitude of their

5 1 τ=.1 fm π xx 1 τ=.1 fm π yy 1 τ=.1 fm π xy τ=3.9 fm τ=3.9 fm -7e- -e- -5e- -e- -3e- -e- τ=3.9 fm Y (fm)..3. Y (fm) Y (fm) τ=7. fm. τ=7. fm -.1 τ=7. fm X (fm) X (fm) X (fm) Figure 3.1: The contour plot of shear stress components in XY plane at three different times. left panel is the evolution of π xx, the middle panel is for π yy, and π x,y is shown in the right panel. The simulation was done for τ π =.5 (3η/p). initial values at τ 7 fm. The π xy (x,y) however shows a non-monotonic variation with τ. Starting from zero it reaches a minimum value around τ 3 fm and finally goes to zero around τ = fm. Simulation with smaller relaxation time for shear stress shows a faster decrease in the magnitude of π xx, π yy, and π xy with time. This is shown in figure 3.. The temporal evolution of the spatial average value of the various shear viscous stress components are shown in the figure 3.3. The solid red line is the simulation for τ π =.5 (3η/p), and the blue dashed curve is the simulation results for τ π =

6 3 1 τ=.1 fm π xx 1 τ=.1 fm π yy 1 τ=.1 fm π xy e-5-5e-5-3e e-5 -e-5 τ=3.9 fm τ=3.9 fm τ=3.9 fm Y (fm) Y (fm)..1. Y (fm) τ=7. fm.5 τ=7. fm τ=7. fm X (fm) X (fm) X (fm) Figure 3.: Same as figure 3.1 but for τ π =.1 (3η/p).

7 π xx π yy e-3 e-3 1e-3 π τx π τy π ττ π xy. 1e-3 5e- -e- -e- -e- 1 1 τ π =.1(3η/p) τ π =.5(3η/p) Au-Au, ε =3 GeV/fm 3 b=7. fm π ηη τ-τ (fm) Figure 3.3: The temporal evolution of spatially averaged components of π µν. The solid red and dashed black curves corresponds to the simulations with τ π =.1 (3η/p) and.5 (3η/p) respectively..1 (3η/p). All the shear stress components except π xy and π ηη remains positive during the whole evolution. Magnitude of all the components rapidly decreases with time and reaches very small value after 7 fm. The rate of decrease is faster for a smaller value of τ π.

8 Au-Au, ε =3 GeV/fm 3 b=7. fm, T f =13 MeV x = y = fm T 3 τ=constant..3 Pb-Pb, ε =1 GeV/fm 3 b=7. fm, T f =13 MeV x = y = fm T 3 τ=constant Temperature (GeV) x = y = fm η/s=. η/s=. η/s=.1 η/s=.1 Temperature (GeV)..3. x = y = fm η/s=. η/s=. η/s=.1 η/s=.1.3 x = y = fm. x = y = fm τ-τ (fm) τ-τ (fm) Figure 3.: The temporal evolution of temperature at three different positions for ideal (red solid), viscous evolution with η/s=. (blue dashed),.1 (green dotted), and.1 (pink dash dotted curve). For comparison the cooling according to the one dimensional Bjorken expansion is also shown (black dashed dot dot curve)for x = y= fm. Left Panel: Simulations for Au-Au collision at b=7. fm with ǫ =3 GeV/fm 3. Right panel: Simulations for Pb-Pb collision at b=7. fm with ǫ =1 GeV/fm 3.

9 3.1. Temperature evolution The temporal evolution of temperature of the fluid at three different spatial positions are shown in figure 3. for Au-Au (left panel) and Pb-Pb (right panel) collisions. The central initial energy density for Au-Au and Pb-Pb collisions are 3 GeV/fm 3 [] and 1 GeV/fm 3 [119] respectively. The solid red, blue dashed, green dotted, and pink dash dotted curves corresponds to η/s= (ideal),.,.1, and.1 respectively. For all the cases, the decrease in temperature with time (τ) suggests that the system cools downwithtime. Atearlytimes( 1fm)therateofcoolingissimilarforonedimensional Bjorken expansion (black dashed dot dot curve) and +1D hydrodynamic expansion. However, after that, the rate of cooling for +1D expansion is slower compared to the one dimensional Bjorken expansion [] for Au-Au collision with ǫ =3 GeV/fm 3. For Pb-Pb collision with a higher ǫ =1 GeV/fm 3, the system cools down faster with transverse expansion compared to Bjorken expansion at later times ( > fm). In +1D hydrodynamics the rate of cooling is slower for a larger η/s for both of the systems considered here. Although the general features as described above remains same at other locations (x = y= fm and x = y= fm) considered here, the magnitude of the initial temperatures decreases with increasing distance from the centre. The rate of cooling also decreases as one goes from central (small x,y) to peripheral (large x,y) regions.

10 7. Au-Au, ε =3 GeV/fm 3 b=7. fm. Pb-Pb, ε =1 GeV/fm 3 b=7. fm <<v T >>.. η/s=. η/s=. η/s=.1 η/s=.1 <<v T >>.. η/s=. η/s=. η/s=.1 η/s= τ-τ (fm) τ-τ (fm) Figure 3.5: The temporal evolution of average transverse velocity v T of the fluid for ideal (red solid), viscous evolution with η/s=. (blue dashed),.1 (green dotted), and.1 (pink dashed doted curve). Left Panel: Simulations for Au-Au collision at b=7. fm with ǫ =3 GeV/fm 3. Right panel: Simulations for Pb-Pb collision at b=7. fm with ǫ =1 GeV/fm Transverse flow and eccentricity We assume the transverse velocity of the fluid at the initial time is zero for our simulation. Because of the pressure gradients, the fluid velocity in the transverse direction gradually builds up with time. The rate of increase in the transverse velocity depends on the speed of sound which in turn depends on the EoS. Figure 3.5 shows the temporal evolution of spatially averaged value of the transverse velocity v T for Au-Au (left panel) and Pb-Pb (right panel) collision with four different values of η/s. The spatially averaged transverse velocity is defined as v T = γ T v x +vy γ T. Here the angular bracket... denotes average with respect to the energy density, and γ T = 1/ 1 v x v y. The red solid, blue dashed, green dotted, and pink dash dotted curves are the simulated v T for η/s= (ideal),.,.1, and.1 respectively. Because of the enhanced pressure in the transverse direction in the viscous fluid, the

11 fluid acceleration is more for shear viscous evolution than in the ideal fluid evolution. This leads to a larger v T for shear evolution compared to the ideal fluid evolution. The increase in v T is more for larger η/s. The v T increases faster with time for simulation with ǫ =1 GeV/fm 3 compared to ǫ =3 GeV/fm 3. In lattice+hrg EoS the speed of sound is larger in the QGP phase than in the hadronic phase. The QGP phase life time is extended for a higher initial temperature or ǫ. This leads to a rapid increase in the velocity for a larger ǫ compared to a fluid evolution with a smaller value of ǫ. The pressure gradient is larger for a higher value of initial ǫ, correspondingly the v T at freezeout is larger for Pb-Pb collision with ǫ =1 GeV/fm 3 compared to Au-Au collision with ǫ =3 GeV/fm 3. The higher values of v T leads to a flatter p T spectra. A non-zero impact parameter collision between two identical nuclei leads to an elliptical collision zone. The spatial eccentricity ε x of the collision zone is defined as ε x = y x y +x. (3.3) ε x is a measure of spatial deformation of the fireball from spherical shape. A zero value of ε x means the system is spherical, < ε x < 1 indicates an elliptic shape with major axis along Y direction, and ε x < means the major axis along X direction. The angular bracket... implies an energy density weighted average. For b=7. fm collision, the evolution of ε x with time (τ) is depicted in figure 3. for (a) Au-Au collision with ǫ =3 GeV/fm 3 and (b) Pb-Pb collision with ǫ =1 GeV/fm 3. Also shown in the same figure are the corresponding momentum anisotropy ε p. Similar to the spatial anisotropy, one can define the asymmetry of fireball in momentum space.

12 9 The momentum space anisotropy ε p is defined as ε p = dxdy(t xx T yy ) dxdy(t xx +T yy ). (3.) The solid red curve corresponds to the temporal evolution of both ε x and ε p for ideal fluid, the dashed, green dotted, and pink dash dotted curves are for shear viscous fluid evolution with η/s =.,.1, and.1 respectively. Because of the enhanced pressure gradient in shear viscous evolution, the initial spatial deformation (ε x. for Au-Au and.3 for Pb-Pb) takes a smaller time to change its shape for the shear viscous evolution compared to the ideal fluid evolution. Because of the higher initial energy density in Pb-Pb collision, the lifetime of the fireball is larger, as well as the pressure gradients, compared to Au-Au collision. Evolution for a longer time with a higher fluid velocity leads to a negative values of ε x at the late stage τ 9 fm for Pb-Pb collision. However, the rate of increase of ε p for a viscous fluid evolution is larger compared to ideal fluid evolution at the early time. ε p saturates at τ fm for shear viscous evolution but continues to grow until the freezeout for ideal evolution. The trend is similar for Au-Au and Pb-Pb collisions except that the rate of increase of ε p is faster for the later case. After τ fm, the ε p becomes smaller for higher values of η/s. The simulated elliptic flow v in hydrodynamic model is directly related to the temporal evolution of the momentum anisotropy. Hence, one should expect the values of v is decreased with increase in η/s.

13 ε x Au-Au, ε =3 GeV/fm 3 b=7. fm η/s=. η/s=. η/s=.1 η/s=.1..3 ε x Pb-Pb, ε =1 GeV/fm 3 b=7. fm η/s=. η/s=. η/s=.1 η/s=.1 ε p,ε x ε p ε p,ε x..1 ε p τ-τ (fm) τ-τ (fm) Figure 3.: The temporal evolution of spatial eccentricity (ε x ) and momentum anisotropy (ε p ) for ideal (red solid), and viscous fluid with η/s=. (blue dashed),.1 (green dotted), and.1 (pink dashed doted curve). Left Panel: Simulations for Au-Au collision at b=7. fm with ǫ =3 GeV/fm 3. Right panel: Simulations for Pb-Pb collision at b=7. fm with ǫ =1 GeV/fm Spectra and Elliptic flow As discussed in chapter-, there are two-fold corrections to the ideal fluid due to shear viscosity. The energy momentum tensor changes due to the dissipative fluxes, and the freezeout distribution function is also modified. If the system is in a state of near local thermal equilibrium then one can calculate the corresponding non-equilibrium correction δf(x,p) to the equilibrium distribution function f eq (x,p) by making a Taylor series expansion of f eq (x,p) [3]. This method will break down for a system which is far away from the state of local thermal equilibrium. In this section we will concentrate on the p T spectra and v of π only for Au-Au collisions (a) with only shear viscous correction in T µν and (b) with both the shear viscous correction to T µν and freezeout distribution function f neq (x,p) = f eq (x,p) + δf shear. We will also investigate about the relative correction to the invariant yield of π due to shear viscosity compared to

14 91 dn/d p T dy (GeV - ) without R Au-Au, ε =3 GeV/fm 3 b=7. fm, T f =13 MeV π -, midrapidity with R dn/d p T dy (GeV - ) η/s=. Au-Au, ε =3 GeV/fm 3 b=7. fm, T f =13 MeV π -, midrapidity η/s=. η/s=.1 η/s=.1 δn/n R-term (%) p T (GeV) δn/n eq (%) p T (GeV) Figure 3.7: Left panel: The p T spectra of π for shear viscous evolution (η/s =.1) with (green dot) and without (solid red curve) the R-term. The bottom panel shows the relative change in p T spectra due to the R-term, where δn = N R term N withoutr term. Rightpanel: Thep T spectraofπ forfourdifferentvaluesofη/s=(redline),.(blue dahsed line),.1 (green dotted line), and.1 (black solid line). The bottom panel shows the relative correction to the p T spectra for shear viscosity compared to the ideal fluid. δn is the difference between shear (N neq ) and ideal (N eq ) p T spectra. the ideal fluid evolution. In addition, we will also discuss the contribution of R-term to shear viscous evolution. The effect of dissipative correction to the freezeout distribution function on the p T spectra and v of π will be presented for two values of T fo =13 and 1 MeV. 3.3 Without correction to the freezeout distribution function Figure 3.7 shows the p T spectra of π in Au-Au collisions with b=7. fm, ǫ =3 GeV/fm 3, and for temperature independent η/s=.1. The top left panel shows the

15 9 results with (green dotted line) and without (solid red line) R-term in the shear viscous evolution. The R-term ensures that throughout the evolution shear stress tensor remains traceless and transverse to the fluid velocity. The relative correction δn = N R term N withoutr term due to the presence of R-term in simplified Israel- Stewart equation to the p T spectra is shown in the bottom left panel of figure 3.7. For p T < 3 GeV the relative correction due to the R-term is <1%. The top right panel of the figure 3.7 shows the p T spectra of π for four different values of η/s=,.,.1, and.1. The spectra becomes flatter with increasing values of η/s. The shear viscous correction to the ideal p T spectra is also p T dependent, the relative correction to the invariant yield (δn/n eq ) for shear viscosity with respect to ideal fluid should be <1. For the current configuration we find that the relative correction remains under 5% for p T values of 1.5, 1., and. GeV for η/s =.,.1, and.1 respectively (shown in the bottom right panel of figure 3.7). The corresponding results for v of π is shown in the figure 3.. We find the effect of R-term in shear evolution on v is negligible (< %). The shear viscosity opposes any anisotropy in the fluid velocity arising due to the pressure gradient. Hence in presence of shear viscosity the momentum eccentricity ε p is reduced (as seen in the left panel of figure 3.) which leads to a reduction in value of v (p T ) (as seen in top right panel of figure 3.). The reduction in v is more for a larger η/s. The relative correction to v due to shear viscosity compared to ideal fluid is within 5% for η/s=.1 for the p T range studied. This is shown in the bottom right panel of figure 3..

16 93 V (p T ) δv /V R-term (%) without R with R Au-Au, ε =3 GeV/fm 3 b=7. fm, T f =13 MeV π -, midrapidity p T (GeV) V (p T ) δv /V eq (%) η/s=. η/s=. η/s=.1 η/s=.1 Au-Au, ε =3 GeV/fm 3 b=7. fm, T f =13 MeV π -, midrapidity p T (GeV) Figure 3.: Left panel: Same as left panel of figure 3.7 but for v. Right panel: Same as right panel of figure 3.7 but for v. 3. Correction to the freezeout distribution function As discussed earlier, there are two-fold corrections to the ideal hydrodynamics due to the dissipative processes. So far all the simulated results shown in the previous sections are for shear viscous evolution with the viscous correction in T µν only. Here we will discuss the effect of shear viscous correction to the freezeout distribution function on p T spectra and v of π. The non-equilibrium correction due to shear viscosity δf shear to the equilibrium freezeout distribution function f eq is calculated from kinetic theory [9, 11]. The distribution function for a system slightly away from local thermal equilibrium can be approximated as [3] f neq (x,p) = f eq (x,p)[1+φ(x,p)], (3.5) where φ(x, p) << 1 is the corresponding deviation from the equilibrium distribution function f eq (x,p). The non-equilibrium correction φ(x,p) can be approximated in

17 9 Grad s 1 moment method by a quadratic function of the four momentum p µ in the following way [11] φ(x,p) = ε ε µ p µ +ε µν p µ p ν, (3.) whereε,ε µ,andε µν arefunctionsofp µ,metrictensorg µν,andthermodynamicvariables. For a system where only the shear stresses exists, one can identify where φ(x,p) = ε µν p µ p ν, (3.7) ε µν = 1 (ǫ+p)t π µν. (3.) As expected the correction factor increases with increasing values of shear stress π µν. The correction term also depends on the particle momentum. The Cooper-Frey formula [1] for a non equilibrium system is [9, 11] dn = d p T dy dn or, d p T dy eq + dn d p T dy neq = + g (π) 3 g dσ (π) 3 µ p µ f neq (p µ u µ,t) g dσ (π) 3 µ p µ f eq (p µ u µ,t) dσ µ p µ δf shear (p µ u µ,t). (3.9) For our case, the product of particle four momentum p µ = (m T coshy,p x,p y,m T sinhy) and the freezeout hypersurface dσ µ = ( m T coshη, τ f x, τ f y m Tsinhη ) τ f dxdydη is expressed as p µ.dσ µ = ( m T cosh(η y) p T. T τ f ) τf dxdydη. Using these relationships into equation 3.9 we have the correction to the invariant yield due to the shear viscosity as dn d p T dy neq = g (π) 3 dσ µ p µ f(x,p)φ(x,p). (3.1) Σ

18 95 After some algebra (see Appendix E) we have the final form of the shear viscous correction to the invariant yield of ideal fluid as Where dn neq dyd p T = g (ǫ+p)t (π) 3 τ f dxdy ( 1) n+1 e nβ[γ p T v T +µ] n=1 ] k (nβ T ) ] [ a1 [m T k 3(nβ T )+ 3a 1 k 1(nβ T )+ a k (nβ T )+ a p T.( [ a1 T τ f ) k (nβ T )+a k 1 (nβ T )+( a 1 +a 3)k (nβ T ) ]. (3.11) a 1 = m T(π ττ +τ π ηη ), a = m T (p x π τx +p y π τy ), (3.1) a 3 = p xπ xx +p yπ yy +p x p y π xy m Tτ π ηη. (3.13). The top left panel of figure 3.9 shows the p T spectra of π with (blue dashed and black solid curve) and without (red solid and green dotted curve) the shear viscous correction to the equilibrium freezeout distribution function for two different freezeout temperatures T fo =13 and 1 MeV respectively. The shear viscous evolution is carried out for η/s =.1. For a lower value of freezeout temperature (T fo =13 MeV), the system evolves for a longer time compared to a higher freezeout temperature (T fo =1 MeV). The shear stresses π µν decrease with time, and for a fluid evolution for longer time, their values on the freezeout hypersurface become vanishingly small (see figure 3.3). Thus depending on the values of the freezeout temperature, one would get different shear viscous correction to the freezeout distribution function. The relative shear viscous correction δn/n wo to the invariant yield of π for T fo =13 MeV and

19 9 dn/d p T dy (GeV - ) 1 3 T f =13 MeV,wo Au-Au, ε =3 GeV/fm 3 b=7. fm π -, midrapidity T f =13 MeV, w T f =1 MeV, wo T f =1 MeV, w v (p T ) T f =13 MeV,wo T f =13 MeV, w T f =1 MeV, wo T f =1 MeV, w Au-Au, ε =3 GeV/fm 3 b=7. fm π -, midrapidity δn/n wo (%) p T (GeV) δv /V wo (%) p T (GeV) Figure3.9: Leftpanel: Thep T spectraofπ forau-aucollisionatb=7.fmwithǫ =3 GeV/fm 3 with and without the shear viscous correction to the equilibrium freezeout distribution function for two different freezeout temperature T f =13 and 1 MeV. The left bottom panel shows the relative correction δn/n wo (see text for details) for T f =13 (black dashed line) and T f =1 (green dotted curve). Right panel: Same as left panel but for v of π. T fo =1 MeV are shown in the bottom left panel of figure 3.9. Where δn = N w N wo ; N w is the invariant yield of π obtained for shear viscous evolution by considering the viscous correction to both T µν and f(x,p), and N wo is the corresponding yield calculated without the viscous correction to the freezeout distribution function but only considering the corresponding viscous correction to T µν. We observe that for freezeout temperature T fo =13 MeV, the non-equilibrium correction dn d p T dy neq to the p T spectra of π due to the δf shear is zero in the p T range considered here. This can be seen from the left bottom panel of figure 3.9, where the relative corrections δn/n wo are shown for T fo =13 MeV (black-dashed line) and T fo =1 MeV (green dotted curve). For T fo =1 MeV, the freezeout correction is nonzero. At low p T the correction is negative and at high p T (> 1.5 GeV) it has a positive

20 97 value. The corresponding effect of the δf shear on v (p T ) of π is shown in the right panel of figure 3.9. The different lines bear the same meaning as used in the left panel. Similar to the p T spectra, the relative correction δv /v wo is close to zero for T fo =13 MeV. However, for the freezeout temperature T fo =1 MeV, this correction is large and is of the order of % at p T 1 GeV.